Yes, many photo filters work equally well on camcorders. Your camcorder's automatic iris and white balance controls may null out the effect you're trying to create with colored and graduated filters, so set these controls manually if possible.

What are some standard filter sizes?

Common sizes are 46, 49, 52, 55, 58, 62, 67, 72 and 77mm.

I want to darken the sky on some of my landscape pictures without changing the brightness of the foreground. Is there a filter that can do this?

Yes, a graduated neutral density filter is clear on one half, gradually becoming gray on the other. It is in a rotating mount, so you can darken one half of the picture without changing the exposure on the remainder.